We have been on several princess cruises, and signing up for slots was never needed, even on sailings with a relatively high number of kids.
My son has autism that was undiagnosed at the time he was in kids club (ages 8-9, pre-covid), so we don't know what it looks like when you can give them a heads up, but I can say he did quite well in Princess kids club but did not do well in the club on another line during that timeframe. Both my kids prefer the princess club to other lines. The club has fun, structured activities. According to my kids, they are expected to participate in the scheduled activity but sometimes an alternative is offered. My son also struggles with being "forced" to do the activity, and he managed ok with having the alternative. I suspect you could also ask for specific alternatives for your son.
